Evolution problems in materials with fading memory

Journal of Nonlinear Mathematical Physics, 2005

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared. The introduction of a Volterra type integral to model a rigid heat conductor in such a way to avoid the infinte speed heat propagation goes back to Cattaneo [3], who suggested a new generalized Fourier's law which linearly relates the heat flux, its time derivative and the temperature-gradient. Subsequently, Coleman's results [4] concerning materials with memory, induced Gurtin and Pipkin [20], to proposed a non-linear model, as well as a linearized version of it, which generalizes the previous ones. Among the many results, since then, in the study of heat transfer phenomena, those more directly of interest in connection to the present approach have been obtained by Gurtin [19] and by Coleman and Dill [5]. They studied properties of free energy functionals in the case of materials with memory and, subsequently, Giorgi and Gentili [16] investigated the heat conduction problem in a fading memory material. The thermodynamical model here adopted is the same studied by Fabrizio, Gentili and Reynolds [11], who considered the thermodynamics of a rigid homogeneous linear heat conductor with

Nonlinear Analysis: Real World Applications

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analyzed; it is obtained coupling an integro-differential equation modeling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modeling the presence of a magnetic field. The case under investigation generalizes a previous study since the relaxation function is allowed to be unbounded at the origin, provided it belongs to L 1 ; the magnetic model equation adopted, as in the previous results [21, 22, 24, 25] is the penalized Ginzburg-Landau magnetic evolution equation.

Evolution Equations and Control Theory, 2014

The existence and uniqueness of solution to an integro-differential problem arising in heat conduction with memory is here considered. Specifically, a singular kernel problem is analyzed in the case of a multi-dimensional rigid heat conductor. The choice to investigate a singular kernel material is suggested by applications to model a wider variety of materials and, in particular, new materials whose heat flux relaxation function may be superiorly unbounded at the initial time t = 0. The present study represents a generalization to higher dimensions of a previous one concerning a 1-dimensional problem in the framework of linear viscoelasticity with memory. Specifically, an existence theorem is here proved when initial homogeneous data are assumed. Indeed, the choice of homogeneous data is needed to obtain the a priori estimate in Section 2 on which the subsequent results, are based.

arXiv (Cornell University), 2016

We consider the model equation arising in the theory of viscoelasticity ∂ tt u − h t (0)∆u − ∞ 0 h ′ t (s)∆u(t − s)ds + f (u) = g. Here, the main feature is that the memory kernel h t (•) depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. [11]. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.

International Journal of Differential Equations …, 2011

Some linear evolution problems arising in the three-dimensional theory of thermoviscoelasticity with memory are considered. Assuming different sets of boundary conditions, the associated evolution systems are formulated in a history space setting. Making use of semigroup techniques, well-posedness results are discussed, as well as asymptotic behavior of solutions. When both mechanical and thermal memory kernels decay exponentially, the longtime behavior of solutions is proved to be of the same type. Under adiabatic boundary conditions, but restricting our attention to a one-dimensional solid, the same decay is proved for a thermoelastic model based on the linearized version of the Gurtin-Pipkin heat conduction law. *

1998

The constitutive law relating stress to strain for viscoelastic materials can be written as a Volterra equation of the second kind. This results in the mathematical models of viscoelastic behaviour taking the form of partial differential equations with memory. In this article we illustrate how the memory terms arise in these equations and also summarize the various partial differential Volterra equations used when modelling problems of quasistatic and dynamic viscoelasticity, and non-Fickian diffusion in polymers. We also indicate some of the numerical analysis work that has been carried out for these problems.

Numerical Algorithms, 2013

Journal of Integral Equations and Applications, 1988

arXiv: Mathematical Physics, 2018

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel inte...

Nonlinear Analysis: Theory, Methods & Applications